What is Loss Aversion?

What is Loss Aversion?
Ulrich Schmidt, Lehrstuhl f¨
ur Finanzmarkttheorie, Universit¨
at Hannover, Germany
Horst Zank,* School of Economic Studies, The University of Manchester, United
Kingdom
Revised Version, 31 August, 2004
Corresponding author: Ulrich Schmidt, Lehrstuhl f¨
ur Finanzmarkttheorie, Universit¨
at Hannover, K¨onigsworther Platz 1, 30167 Hannover, Germany; Telephone:
++49 511 7625117, Fax: ++49 511 7625240, E-mail: [email protected]
1
Abstract. A behavioral definition of loss aversion is proposed and its implications for original and cumulative prospect theory is analyzed. Original prospect
theory is in agreement with the new loss aversion condition, and there utility is
capturing all effects of loss aversion. In cumulative prospect theory loss aversion is
captured by both, the weighting functions and the utility function. Further, some
restrictions apply for the weighting functions involved in the latter model.
Keywords: prospect theory, loss aversion
Journal of Economic Literature Classification: D81, C60
2
Loss aversion is an important psychological concept which receives increasing attention in economic analysis. It has first been proposed by Kahneman and Tversky
(1979) in the framework of prospect theory, and later it has also been defined for
choice under certainty by Tversky and Kahneman (1991). The popularity of loss
aversion is due to the fact that it can explain many phenomena which remain paradoxes in traditional choice theory. Well-known examples are the endowment effect
(Thaler, 1980), the equity premium puzzle (Benartzi and Thaler, 1995), and the
status quo bias (Samuelson and Zeckhauser, 1988). In recent years loss aversion has
also frequently been applied in behavioral finance (cf. Barberis et al., 2001; Barberis
and Huang, 2001; Berkelaar and Kouwenberg, 2000a,b; Roger, 2003; and Gomes,
2003). A further important aspect of loss aversion is the fact that it can resolve the
criticism on expected utility put forward by Rabin (2000, p. 1288) and Rabin and
Thaler (2001) who showed that reasonable degrees of risk aversion for small and
moderate stakes imply unreasonable high degrees of risk aversion for large stakes.
Kahneman and Tversky’s (1979, p. 279) view of loss aversion is as follows: An
individual is loss averse if she or he dislikes symmetric 50-50 bets and, moreover,
the aversiveness to such bets increases with the absolute size of the stakes. This
clearly is a behavioral concept defined entirely in terms of preferences. As such, the
concept is model independent. Kahneman and Tversky (1979) showed that, in the
framework of prospect theory, this definition of loss aversion is equivalent to a utility
function which is steeper for losses than for gains. As probability weighting played
no role in the derivation of this result, it appears that the effect of loss aversion is
captured solely by the utility function. It is, therefore, not surprising that nearly all
work on loss aversion employed utility as the carrier of loss aversion. For instance,
Tversky and Kahneman (1992, page 303) assume that utility is steeper for losses
than for gains. Wakker and Tversky (1993) propose a preference condition based on
a cardinal utility index independent of probability weighting. The latter condition
has empirically been confirmed in a recent test in Schmidt and Traub (2002). In
a review of non-expected utility theories Starmer (2000) highlights the descriptive
3
advantages of rank and sign dependent models, and summarizes loss aversion as
utility being steeper for losses than for gains. Benartzi and Thaler (1995) view
loss aversion as a property of utility exhibited at the status quo. This view is
also adopted in K¨obberling and Wakker (2003), where an index of loss aversion is
defined as the ratio of the left and right derivative of utility at the status quo. All
the previous conditions employ a comparison of utility differences between gains and
losses of equal absolute size. In contrast Neilson (2002) suggests stronger conditions
by dropping this symmetry requirement.
It can be concluded that, the way loss aversion is currently understood, it is
essential to identify utility independent of probability weights prior to any analysis
of loss attitudes. Moreover, given that most definitions of loss aversion are not
formulated in terms of preferences, it follows that loss aversion is no longer model
independent. Since most studies mentioned above do not use original prospect
theory but the modern cumulative prospect theory instead, their notion of loss
aversion does no longer agree with the behavioral concept proposed by Kahneman
and Tversky (1979).
The goal of this note is to analyze loss aversion in more detail, first by showing
that the current definitions of loss aversion have different behavioral implications
for different decision models, and second by providing model independent definitions of loss aversion. By taking the view that loss aversion is solely an aspect of
gain/loss comparisons, and therefore independent of the ranking of outcomes and
the likelihood of occurrence, we propose a preference condition which empirically
can easily be verified. We then show that, for original prospect theory, this condition is equivalent to the condition proposed by Kahneman and Tversky (1979).
For original prospect theory probability weighting has no influence on loss attitudes,
and consequently utility being steeper for losses than for gains is equivalent to loss
aversion as defined by the new condition. However, when the modern version of
prospect theory of Tversky and Kahneman (1992) is assumed, the new condition
implies some surprising restrictions for the representing functional. Now utility be-
4
ing steeper for losses than for gains is neither necessary nor sufficient for loss averse
behavior. Rather it is the case that the ratio between the utility difference for losses
and the utility difference of corresponding gains must exceed the ratio between the
gain decision weight and the loss decision weight generated by any common probability. The special case of rank-dependent utility preferences is also discussed, and it is
shown that when removing the status quo effects from our behavioral definition, loss
aversion comes down to the well-known aversion to mean preserving spreads. Before
we provide details of the analysis in Section 2, we introduce the formal framework
in the next section.
1
Notation and Definitions
A lottery is a finite probability distribution over the set of monetary outcomes
(here identified with the set of real numbers, IR).
It is represented by P :=
(p1 , x1 ; . . . ; pn , xn ) meaning that probability pj is assigned to outcome xj , for j =
1, . . . , n. The probabilities pj are nonnegative and sum to one. With this notation we implicitly assume that outcomes are ranked in decreasing order, i.e.,
x1 > · · · > xk > 0 > xk+1 > · · · > xn for some 0 6 k 6 n. Positive outcomes
are called gains and negative outcomes are losses.
Expected Utility (EU) holds if an individual evaluates lotteries according to:
EU (p1 , x1 ; . . . ; pn , xn ) =
n
X
pi U(xi ),
(1)
i=1
where U is the utility function. In this paper U is assumed strictly increasing and
continuous with U (0) = 0. In contrast to the traditional interpretation we assume
that U is defined on gains and losses and not on final wealth positions. This allows
us to analyze loss attitudes in the EU model.
In the original version of prospect theory (OPT) there exists a weighting function
w (i.e., w : [0, 1] → [0, 1], strictly increasing with w(0) = 0 and w(1) = 1), such that
5
lotteries are evaluated by
OP T (p1 , x1 ; . . . ; pn , xn ) =
n
X
w(pi )U(xi ).
(2)
i=1
This formula applies only to lotteries which involve both, gain and loss outcomes.
Actually, Kahneman and Tversky (1979) proposed OPT only for lotteries with three
outcomes of which one equals zero. Here we use the extension to general lotteries
(for instance as used by Fennema and Wakker, 1997). Recall that under OPT
the weights involved in the evaluation of a lottery are direct transformations of the
probabilities, an aspect that was criticized because it leads to violations of stochastic
dominance (e.g., Fishburn, 1978).
The modern version of prospect theory, called cumulative prospect theory (CPT),
differs from OPT by involving two weighting functions w+ , w− , and moreover, the
weights used in the evaluation of lotteries are differences in decumulative, respectively, cumulative probabilities. All lotteries are evaluated by
k
X
CP T (p1 , x1 ; . . . ; pn , xn ) =
[w+ (p1 + · · · + pi ) − w+ (p1 + · · · + pi−1 )]U(xi ) (3)
i=1
+
n
X
[w− (pi + · · · + pn ) − w− (pi+1 + · · · + pn )]U(xi ).
i=k+1
For the special case that w+ equals the dual of w− (i.e., w+ (p) = 1 − w− (1 − p)
for 0 6 p 6 1), the cumulative prospect theory functional comes down to the rank-
dependent utility (RDU) functional:
RDU (p1 , x1 ; . . . ; pn , xn ) =
n
X
[w(p1 + · · · + pi ) − w(p1 + · · · + pi−1 )]U (xi ).
(4)
i=1
Note, however, that, similar to EU, general RDU does not require utility to be
defined over changes in wealth.
2
Analysis
We start the analysis in this section by contrasting the development of the concept
of loss aversion with the development and understanding of the concept of risk
6
aversion. Most of the empirically observed risk aversion is driven by loss aversion,
and because loss attitude seems to be an intrinsic component of risk attitudes it
is expected that the development of these concepts is analogous. According to
Rothschild and Stiglitz (1970), strong risk aversion holds if and only if an individual
always dislikes mean-preserving spreads in risk. This is a behavioral concept of risk
aversion since it is defined in terms of preferences. It is well-known that this concept
of risk aversion is equivalent to the concavity of the utility function in expected
utility theory. Under that theory weak risk aversion (i.e., preference for expected
value) is also equivalent to a concave utility. Nevertheless, the subsequent literature
kept on defining risk aversion in terms of the behavioral concept (e.g., aversion to
mean-preserving spreads, weak risk aversion, aversion to monotone mean-preserving
spreads (Quiggin, 1991), etc.) and not in terms of properties of the functional
representation (e.g., concavity of the utility function).
It is important to make the distinction between choice behavior and properties of
utility because in alternative theories the implications for utility may vary. For example, in rank-dependent utility a concave utility function is consistent with strong
risk averse behavior (see Chew, Karni, and Safra, 1987), however, weak risk aversion
can hold if utility is convex (Chateauneuf and Cohen, 1994). Recently, Chateauneuf,
Cohen and Meilijson (2004a) have demonstrated that, under rank-dependent utility, aversion to monotone mean preserving spreads holds if and only if the index
of pessimism, which depends solely on the weighting function, exceeds the degree
of non-concavity of the utility function, called index of greediness. Hence, nonconcave utility may also coexist with aversion to monotone mean preserving spreads
(see also Chateauneuf, Cohen and Meilijson, 2004b, and Abouda and Chateauneuf,
2002). Assuming CPT, Schmidt and Zank (2002), show that strong risk aversion
can hold if utility is concave for losses, concave for gains, but convex at the status
quo. Therefore, identifying risk aversion with a concave utility means that expected
utility is implicitly assumed.
A striking similarity holds for loss aversion: by identifying loss aversion with
7
a utility that is steeper for losses than for gains, one implicitly assumes original
prospect theory and not the modern version of cumulative prospect theory (see
Propositions 2 and 4 below). To clarify on this point let us now turn to Kahneman
and Tversky’s (1979) interpretation of loss aversion: individuals dislike symmetric
50-50 bets, and additionally, such bets with higher amounts of prizes are dispreferred.
More formally, we state the following definition of loss aversion.
Definition 1 Loss aversion holds if (0.5, x; 0.5, −x) ≺ (0.5, y; 0.5, −y) for all x >
y > 0.
As noted above, this is clearly a behavioral concept. It turns out that in the
framework of prospect theory or expected utility theory (that is OPT with a linear weighting function for probabilities) loss aversion is equivalent to the fact that
the utility function is steeper for losses than for gains, such that we may have a
kink at the status quo. This result follows immediately from substitution of the
corresponding functional form, and is summarized in Proposition 2.
Proposition 2 In EU and OPT loss aversion is satisfied if and only if for all
x > y > 0 it holds that U (x) − U(y) < U(−y) − U(−x).
¤
Note that in the definition of loss aversion the restriction to 50-50 symmetric
bets is rather arbitrary. Assuming OPT, one can equivalently define loss aversion
as aversion to symmetric bets, where the aversion becomes more pronounced if the
size of the stakes increases. That is, (p, x; 1 − 2p, 0; p, −x) ≺ (p, y; 1 − 2p, 0; p, −y)
for all x > y > 0 and any p ∈ (0, 1/2]. The middle payoff in these lotteries does not
need to be equal to the status quo, and could be any common outcome. This latter
condition has been confirmed in a recent empirical study by Brooks and Zank (2004).
The reason we use this simple variation of the definition is to use lotteries that have
at most two nonzero outcomes, similar to the framework of OPT in Kahneman and
Tversky (1979). Obviously, the size and therefore the rank of the common outcome
in the previous preference has no effect on the shape of the utility function. When
8
assuming this latter condition in Proposition 2, the same implications result for
utility.
Let us now assume that loss attitudes are independent of the rank of outcomes.
Then the following definition of loss aversion is acceptable.
Definition 3 Strong loss aversion holds if for all x > y > 0 and 0 < α 6 0.5, we
have
(p1 , z1 ; ...; pi−1 , zi−1 ; α, x; pi+1 , zi+1 ; ...; pj−1 , zj−1 ; α, −x; pj+1 , zj+1 ; ...; pn , zn )
≺ (p1 , z1 ; ...; pi−1 , zi−1 ; α, y; pi+1 , zi+1 ; ...; pj−1 , zj−1 ; α, −y; pj+1 , zj+1 ; ...; pn , zn ).
The condition says that all that matters for the preference in Definition 3 is that
the absolute size of the payoffs with likelihood α are smaller in the second lottery.
This condition demands that among two lotteries, for which one can win or lose a
given amount with equal probability, that lottery will be preferred for which this
amount is smaller. The preference is independent of all other common outcomes,
however, with the notation used in this paper, the condition presupposes that the
transition from x to y in the second lottery does not change the rank-ordering of
outcomes. Note that loss aversion follows from strong loss aversion by requiring
α = 1/2. If loss aversion is accepted then, intuitively, we do not see any obvious
reason for objecting against strong loss aversion. The motivation for strong loss
aversion stems from the common belief that loss aversion is reflected only through
the utility function. Taking that view, means that the common rank of the outcomes
x, y, respectively, −x, −y, in the lotteries above should not matter, neither should
the magnitude of the common probability α matter.
A closer look at the lotteries in Definition 3 reveals its relationship with aversion
to mean preserving spreads. If we define X and Y as
X = (p1 , z1 ; ...; pi−1 , zi−1 ; α, x; pi+1 , zi+1 ; ...; pj−1 , zj−1 ; α, −x; pj+1 , zj+1 ; ...; pn , zn ),
Y
= (p1 , z1 ; ...; pi−1 , zi−1 ; α, y; pi+1 , zi+1 ; ...; pj−1 , zj−1 ; α, −y; pj+1 , zj+1 ; ...; pn , zn ),
and, by keeping the probability distribution fixed, defining the (not rank-ordered)
9
lottery Z as
Z = (p1 , 0; ...; pi−1 , 0; α, x − y; pi+1 , 0; ...; pj−1 , 0; α, −x + y; pj+1 , 0; ...; pn , 0),
we can write X = Y ⊕ Z, where the symbol ⊕ means that outcomes are added while
keeping the probability distribution fixed. Hence, X results from Y by subtracting
(x −y) from −y and adding this amount to y. Obviously, Z has zero expected value.
Note, that not all such Z are permitted under strong loss aversion, as the nonzero
outcomes in Z need to be added to corresponding outcomes in Y that are of the
same sign. Also, the symmetry in the magnitude of the outcomes (i.e., y versus −y
in Y and x versus −x in X) needs to be maintained after adding outcomes of Z
to those of Y . Strong loss aversion, therefore, does not have any implications for
the behavior on the gain domain or the behavior on the loss domain. Permitting
the addition of outcomes for general lotteries Z, where the only restriction is that
the expected value of Z is zero, which comes down to removing the status quo
effects from the definition of strong loss aversion, gives a stronger condition, namely
aversion to mean preserving spreads. That strong loss aversion is much weaker than
aversion to mean preserving spreads can be inferred from the fact that even under
RDU it implies concavity only locally at 0 (see the remarks following Proposition 7
below). Clearly, removing the status quo effects from the definition of strong loss
aversion implies that utility under RDU is concave everywhere, in agreement with
the results of Chew, Karni and Safra (1987) and Schmidt and Zank (2002).
The next proposition shows that, under OPT (and EU) strong loss aversion has
the same implications as loss aversion. Its proof follows immediate from substituting
the corresponding functional forms and cancelling the common terms.
Proposition 4 In EU and OPT strong loss aversion is satisfied if and only if for
all x > y > 0 it holds that U (x) − U(y) < U(−y) − U (−x).
¤
As pointed out by Starmer (2000) the rank and sign dependent model of CPT
combines mathematical tractability and descriptive validity to accommodate most
stylized facts from experimental studies. Rank-ordering of outcomes did not play
10
any role in OPT, and it is not surprising that it does not play any role in the result
of Propositions 2 and 4. It is probable that by retaining the view that loss aversion
is equivalent to a utility that is steeper for losses than for gains, the important
development initiated by Quiggin (1982) regarding the ranking of outcomes for the
derivation of decision weights has been neglected in the analysis of loss attitudes.
This is in contrast to the concept of risk aversion because the subsequent literature
did not retain the behavioral definition (aversion to symmetric 50-50 bets) of loss
aversion but uses the definition in terms of properties of the utility function. In
fact, all recent formal studies concerned with loss aversion we are aware of define
loss aversion in terms of the shape of the utility function. In our view this is,
however, problematic since most of these papers do not rely on EU or OPT but on
CPT.
In the latter model a steeper utility for gains than for losses has rather different
behavioral implications than in OPT or EU. To highlight this fact the result of the
following proposition, combined with specific CPT preferences shows that a utility
function that is steeper for losses than for gains can accommodate loss seeking (i.e.,
for all x > y > 0 we have (0.5, x; 0.5, −x) Â (0.5, y; 0.5, −y)).
Proposition 5 In CPT loss aversion is satisfied if and only if for all x > y > 0 it
holds that
U(−y) − U(−x)
w+ (0.5)
> −
.
U(x) − U(y)
w (0.5)
(5)
¤
Proposition 5 shows that under CPT utility being steeper for losses than for gains
is neither necessary nor sufficient for loss aversion. By considering the parametric
weighting functions fitted by Tversky and Kahneman (1992), we get w+ (0.5)/w− (0.5) =
0.927, that is, we may have loss aversion even if utility for gains is steeper than utility for losses. Recent non-parametric estimations of weighting functions for CPT
indicate that w− is above w+ away from the extreme probabilities (e.g., Abdellaoui,
2000; Abdellaoui, Vossmann, and Weber, 2003). This suggests that loss aversion
11
may be dictated by the shape of the weighting functions rather than that of utility.
In the case of general CPT preferences where w+ (0.5)/w− (0.5) > 1, we may therefore observe loss seeking behavior if utility is such that the inequality in Proposition
5 is reversed:
Corollary 6 In CPT loss seeking can hold even if we have U (x)−U (y) < U (−y)−
U(−x) for all x > y > 0.
¤
Note that Corollary 6 also holds if rank-dependent utility is assumed. Thus, if
loss aversion is defined in terms of the slope of the utility function, it has different
behavioral implications in different choice theories. A further evidence of this fact
is the work of Wakker and Tversky (1993) who develop a behavioral condition
which implies a steeper utility for losses than for gains in the framework of CPT.
Although this condition differs from the behavioral condition of Kahneman and
Tversky (1979), it has nevertheless been termed loss aversion.
Loss aversion captured solely in the curvature of utility cannot be a useful concept
for decision analysis if it has different behavioral implications in different choice
theories. Therefore, concerning the definition of loss aversion the literature should
agree on a behavioral concept defined in terms of preferences. Clearly, the choice of
common probability p = 0.5 in the behavioral condition of Kahneman and Tversky
is motivated by a common intuition regarding loss aversion, and that any behavioral
definition of loss aversion will need to allow for more flexibility in the choice of such
probabilities. Strong loss aversion does exactly that, and as indicated in Proposition
4, for OPT the implications for utility are identical as those in Kahneman and
Tversky (1979) (see also Proposition 2 above).
Let us now analyze the implication of strong loss aversion for CPT preferences.
The following result can be derived from substitution of CPT for the preference in
Definition 3.
Proposition 7 Under CPT strong loss aversion is satisfied if and only if for all
12
x > y > 0, all 0 < α 6 0.5, and all γ, δ > 0 with 1 − 2α > γ + δ it holds that
w+ (γ + α) − w+ (γ)
U (−y) − U (−x)
> −
.
U (x) − U(y)
w (δ + α) − w− (δ)
(6)
¤
By keeping x and y fixed, inequality (6), rewritten as
[w− (δ + α) − w− (δ)][U(−y) − U(−x)]
> 1,
[w+ (γ + α) − w+ (γ)][U (x) − U(y)]
clarifies on the consequences of strong loss aversion: the impact of increasing a gain
of given size with a given probability by a fixed amount is less than the impact
of decreasing a loss of the same size and with the same probability by the same
amount. Note, that here impact means the combined effect of decision weight and
utility difference. This result supports the view that under CPT loss aversion is
captured by both, probability weighting and utility.
The condition in Proposition 7 has strong implications for the shapes of the
weighting functions. First, if the gain weighting function is too steep near impossibility (as in the usually proposed inverse S-shape) then the right hand side of
inequality (6) can become unreasonably high. For example, in the parametric forms
proposed by Lattimore, Baker, and Witte (1992), Tversky and Kahneman (1992),
and Prelec (1998), this ratio converges to infinity if γ + α converges to 0, as the
right derivative of w+ at impossibility becomes infinity. Therefore, in order to be
compatible with strong loss aversion, the weighting function for gains cannot be “too
steep” over a given range while, using similar arguments, the weighting function for
losses cannot be “too flat.”
A second implication of strong loss aversion is that the weighting function w+
is continuous at impossibility if there exists some 0 < δ < 1 such that w−0 (δ) is a
positive constant.1 Assuming that this were not the case, let (qj )∞
j=1 be a decreasing
sequence of probabilities converging to 0, such that w+ (qj ) is converging to q∗ > 0.
Then, [w+ (qj ) − w+ (0)]/qj → ∞ for j → ∞, and [w− (δ + qj ) − w− (δ)]/qj → w−0 (δ).
This implies that the ratio between the decision weights in inequality (6) becomes
13
infinity, contradicting the assumption of boundedness for the utility function in
CPT. Consequently, w+ must be continuous at 0.
Given these implications one immediate question that arises is regarding an
upper bound for the ratio of decision weights in inequality (6). For general weighting
functions this remains an open question, however, we can give a partial answer to
this question, for the special case of continuous and differentiable inverse S-shaped
weighting functions, which initially are concave and then convex. In that case the
right and left derivative of the weighting functions are well defined. Recall that,
under the assumptions of CPT, the right derivative of w+ at 0 and the left derivative
at 1 must be finite, otherwise inequality (6) is violated. Similarly, w− cannot be too
flat (i.e., w−0 (δ) = 0 for some 0 < δ < 1). For this case of weighting functions the
supremum of the right hand side of (6) can take two forms, depending on whether
w+ is steepest at 0 (and then w− is flattest at its inflection point) or w+ is steeper
for larger probabilities relative to w− for probabilities below its inflection point. So,
either the supremum is equal to the ratio w+0 (0+ )/w−0 (p∗ ), where p∗ denotes the
inflection point of w− , or it is equal to:
sup
06p61
w+0 (p)
.
w−0 (1 − p)
(7)
If these ratios are larger or equal 1 (for example if both weighting functions intersect
the 45-degree line from above) then we can retain the statement that utility is steeper
for losses than for gains.
A further interesting special case is the following one. Assume that the weighting
functions are continuous on (0, 1) and that they intersect in the open interval (0, 1).
Then, there exist some probability p ∈ (0, 1) where the weighting function for gains
is steeper than that for losses, implying that utility must be steeper for losses than
for gains. In fact if both weighting functions are continuous on [0, 1], then w+ must
be steeper than w− at some probability p. For noncontinuous weighting functions
this must not hold, and utility can be flatter for losses than for gains (e.g., if utility
is linear for gains and linear for losses, and the weighting functions are linear and
discontinuous at 1, then one can choose utility steeper for gains than for losses and
14
simultaneously w− sufficiently steeper than w+ such that inequality (6) above holds).
This shows that under reasonable assumptions for the weighting functions utility is
required to be steeper for losses than for gains. In particular this shows that under
RDU with a continuous weighting function strong loss aversion implies that utility
is concave at 0. Whether the degree of loss aversion measured from utility (i.e.,
the left hand side of inequality (6)) is driven by the corresponding degree measured
from the weighting functions (that is the right hand side of inequality (6)) seems
to be an empirical question. Strong loss aversion does not imply that the infimum
of ratios of symmetric utility differences is equal to the supremum of the ratios of
decision weights.
This paper has highlighted the need for a definition of loss aversion in terms
of preference conditions, which represents model independent choice behavior. We
have introduced the concept of strong loss aversion, and stressed some of its implication for the weighting functions in CPT. We have demonstrated that probability
perception plays a key role in the analysis of loss attitudes alongside utility. Whether
strong loss aversion is the most appropriate definition of loss aversion can be debated, and empirical tests of the condition can help clarifying this point. We hope
that these results generate further theoretical analyses on the important issue of loss
attitudes, leading to a satisfactory answer to the question of what is an appropriate
model free definition of loss aversion.
15
Notes:
* We are indebted to Mich`ele Cohen and Peter Wakker for helpful comments.
The suggestions of an anonymous referee have improved the presentation of the
paper.
1. Note that the weighting functions considered here are strictly increasing on
[0, 1], and therefore the derivatives exist almost everywhere. To derive continuity of
w+ at 0 we need to ensure that the derivative of w− is not everywhere equal to 0.
16
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